The method of successive approximation applied to find the probability for an insurance company with random premiums

A risk process that describes the evolution of the capital of an insurance company is analyzed, random premiums and claims being available. Integral equations of nonbankruptcy probability as a function of the initial capital are derived. Necessary and sufficient conditions for the existence and uniqueness of solutions of these integral equations, and convergence conditions for the method of successive approximation for finding their solutions are established __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 112–127, January–February 2006.     

Solving systems of linear equations with Boolean variables

The paper presents a method to solve systems of linear equations with Boolean variables, which implements an enumeration strategy. Necessary and sufficient conditions for the existence of feasible plans are formalized. A formal procedure to analyze subsets of alternatives is described. The structure of an algorithm that possesses the property of completeness is presented. Special cases of systems of equations are examined. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 42–50, September–October 2006.     

Analysis of systems of singular ordinary differential equations

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones. Translated from Kibemetika i Sistemnyi Analiz, No. 5, pp. 103–110, September–October, 1999.     

An optimal control model for a system of degenerate parabolic integro-differential equations

An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 90–102, November–December 2007.     

A model of optimal control over a nonlinear multidimensional innovation diffusion process

A model of competitive innovation diffusion is considered. The model is based on the Lotka-Volterra system and an initial-boundary problem for a system of quasilinear parabolic equations. The maximum principle is proved for the problem of diffusion of two competitive innovations, and sufficient conditions of existence of optimum control are obtained for the system. A numerical algorithm is constructed for solving optimum control problems, and numerical results for a model example are presented. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 120–133, July–August 2008.     

Multiplicative stability criteria based on difference solutions of ordinary differential equations

For computer analysis of Lyapunov stability, multiplicative criteria are proposed that are based on difference approximations to solutions of the Cauchy problem. These criteria can be applied to ordinary differential equations in normal form and include the necessary and sufficient stability conditions. For a system of linear equations with constant coefficients, information on the characteristic polynomial of the coefficient matrix and its roots is not used. The stability analysis is combined with difference solution and simulation of error accumulation. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 127–142, January–February 2006.     

Necessary and sufficient conditions of existence and uniqueness of solutions to integral equations of actuarial mathematics

Necessary and sufficient conditions of existence and uniqueness of solutions of integral equations are established for ruin probability as a function of the initial capital of an insurance company. Several sufficient conditions and correctness conditions of the problem of finding ruin probability are also established. A general method of successive approximations for solution of this problem is substantiated. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 157–164, September–October 2006.     

Methods of solution and criteria of consistency of systems of linear diophantine equations over the set of natural numbers1

Some methods and algorithms of solution of systems of linear Diophantine equations over the naturals are briefly reviewed. Criteria and an incremental algorithm of efficient solution of the problem of consistency for a system of linear Diophantine equations and inequalities over the naturals are given. This research was supported under grant INTAS-RFBR 95-0095. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 12–36, July–August, 1999.     

<Emphasis Type="Italic">G</Emphasis>*-Convergence of Nonlinear Operators in the Theory of Homogenization of Control Objects

To study problems of homogenization of one class of control objects described by nonlinear operator equations, the concept of G*-convergence of nonlinear operators is introduced. The sufficient conditions of identifiability of homogenized objects are obtained. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 42–65, September–October 2005.     

Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space,disturbed by Gaussian processes. Part I

Nonlinear evolution differential equations with unbounded linear operators of disturbance by Gaussian random processes are considered in an abstract Hilbert space. For the Cauchy problem for the differential equations, the sufficient existence and uniqueness conditions for their solutions are proved and the sufficient conditions for the equivalence of the probability measures generated by these solutions are derived. Moreover, the corresponding Radon–Nikodym densities are calculated explicitly in terms of the coefficients or characteristics of the considered differential equations.     

Analysis of stability of stochastic dynamic systems under Poisson perturbations. II. Stability of solutions in quadratic mean of systems of linear stochastic differential equations under Poisson perturbations

Necessary and sufficient conditions of asymptotic stability in quadratic mean are obtained for trivial solutions of systems of linear stochastic differential equations under Poisson perturbations. Model problems are analyzed. Part I of this article is published in No. 4 (2005). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 50–66, November–December 2005.     

Linear systems of first order ordinary differential equations: fuzzy initial conditions

 We present two types of fuzzy solutions to linear systems of first order differential equations having fuzzy initial conditions. The first solution, called the extension principle solution, fuzzifies the crisp solution and then checks to see if its α-cuts satisfy the differential equations. The second solution, called the classical solution, solves the fuzzified differential equations and then checks to see if the solution always defines a fuzzy number. Three applications are presented: (1) predator–prey models; (2) the spread of infectious diseases; and (3) modeling an arms race.     

Direct methods of solving systems of linear algebraic ef ions with complex a-matrices

Algorithms of computer algebra are proposed for solving systems of linear algebraic equations with complex á- matrices. An analysis of roundoff errors for the computational schemes considered is given. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 144–156, March–April, 2000.     

Solution of systems of linear algebraic equations with preliminary flattening

A method of improving computing properties of matrices of systems of linear algebraic equations is considered. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 144–149, September–October, 1999.     

Optimal control models for interregional migration under social risks

An organizational two-level hierarchic system is used to examine control models for interregional migration under conditions of social risks. The Cauchy problem is used to solve the multidimensional transport equation (Problem A1) and a system of ordinary linear differential equations (Problem A2). For the models proposed, sufficient conditions for the existence of optimal control on the classes of functions designed are established, and stable numeral methods of searching for optimum solutions are developed. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 107–122, May–June 2006.     

Nonminimal Kane's Equations of Motion for Multibody Dynamical Systems Subject to Nonlinear Nonholonomic Constraints

Nonholonomic constraint equations that are nonlinear in velocities are incorporated with Kane's dynamical equations by utilizing the acceleration form of constraints, resulting in Kane's nonminimal equations of motion, i.e. the equations that involve the full set of generalized accelerations. Together with the kinematical differential equations, these equations form a state-space model that is full-order, separated in the derivatives of the states, and involves no Lagrange multipliers. The method is illustrated by using it to obtain nonminimal equations of motion for the classical Appell–Hamel problem when the constraints are modeled as nonlinear in the velocities. It is shown that this fictitious nonlinearity has a predominant effect on the numerical stability of the dynamical equations, and hence it is possible to use it for improving the accuracy of simulations. Another issue is the dynamics of constraint violations caused by integration errors due to enforcing a differentiated form of the constraint equations. To solve this problem, the acceleration form of the constraint equations is augmented with constraint stabilization terms before using it with the dynamical equations. The procedure is illustrated by stabilizing the constraint equations for a holonomically constrained particle in the gravitational field.     

Analysis of the properties of a linear system using the method of artificial basis matrices

Formulas relating elements of the method in adjacent basis matrices are used to solve a system of linear algebraic equations and to represent analytically the general solutions to the corresponding system of linear algebraic inequalities for a nondegenerate constraint matrix. Sponsored by the ICS NATO program of April 18 2006, in line with the Project “Optimal replacement of information technologies and stable development (in Kazakhstan, Ukraine, and the USA),” NATO Grant CLG 982209. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 119–127, July–August 2007.     

On solving systems of ordinary differential equations on MIMD-computers

The error of solution of Cauchy problems for systems of ordinary differential equations is estimated in the case where the input data are approximate. It is shown how to prepare a program for computing the right-hand sides of the system automatically and simultaneously. Diagrams are presented to illustrate the efficiency of parallelization. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 175–182, March–April 2007.     

Adaptive diagnostic system based on fuzzy relations

Causes (diagnoses) are retrieved and identified using observed effects (symptoms) based on fuzzy relations and Zadeh’s compositional rule of inference. An approach to designing adaptive fuzzy diagnostic systems is proposed. It allows solving fuzzy logic equations and designing and adjusting fuzzy relations using expert and experimental information. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 135–150, July–August 2009.     

Remarks on stability of solutions of the property domination problem

A class of combinatorial problems is considered whose investigation and solution require the notions of the theory of fuzzy sets. The necessary and sufficient conditions of stability are given. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 36–40, July–August, 1999.